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3
China Western Mathematical Olympiad 2015 ,Problem 3
China Western Mathematical Olympiad 2015 ,Problem 3
Source: China Yinchuan Aug 16, 2015
August 16, 2015
inequalities
Problem Statement
Let the integer
n
≥
2
n \ge 2
n
≥
2
, and
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots,x_n
x
1
,
x
2
,
⋯
,
x
n
be positive real numbers such that
∑
i
=
1
n
x
i
=
1
\sum_{i=1}^nx_i=1
∑
i
=
1
n
x
i
=
1
.Prove that
(
∑
i
=
1
n
1
1
−
x
i
)
(
∑
1
≤
i
<
j
≤
n
x
i
x
j
)
≤
n
2
.
\left(\sum_{i=1}^n\frac{1}{1-x_i}\right)\left(\sum_{1\le i<j\le n} x_ix_j\right)\le \frac{n}{2}.
(
i
=
1
∑
n
1
−
x
i
1
)
(
1
≤
i
<
j
≤
n
∑
x
i
x
j
)
≤
2
n
.
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